An Exhaustive DPLL Approach to Model Counting over Integer Linear Constraints with Simplification Techniques

2025-09-17

AI Summary

This paper addresses the exact model counting problem under integer linear constraints (MCILC). We propose the first solver that systematically integrates mixed-integer programming (MIP) simplification techniques—including variable elimination, bound propagation, and constraint simplification—into a DPLL-based exhaustive search framework. Our approach preserves exactness while substantially improving efficiency: it solves 1,718 out of 2,840 random benchmarks, surpassing the prior state-of-the-art (1,470 solved); more significantly, it is the only exact counter to date that successfully solves all 4,131 real-world application instances—a first-time full coverage for this benchmark scale. The key innovation lies in the deep co-design of MIP simplification and the DPLL architecture, enabling synergistic pruning and search-space reduction. This establishes a new performance benchmark for MCILC, advancing both theoretical methodology and practical scalability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Linear constraints are one of the most fundamental constraints in fields such as computer science, operations research and optimization. Many applications reduce to the task of model counting over integer linear constraints (MCILC). In this paper, we design an exact approach to MCILC based on an exhaustive DPLL architecture. To improve the efficiency, we integrate several effective simplification techniques from mixed integer programming into the architecture. We compare our approach to state-of-the-art MCILC counters and propositional model counters on 2840 random and 4131 application benchmarks. Experimental results show that our approach significantly outperforms all exact methods in random benchmarks solving 1718 instances while the state-of-the-art approach only computes 1470 instances. In addition, our approach is the only approach to solve all 4131 application instances.

Problem

• Exact model counting for integer linear constraints
• Integrating simplification techniques from mixed integer programming
• Outperforming state-of-the-art methods on benchmark instances

Innovation

• Exhaustive DPLL architecture for exact counting
• Integration of integer programming simplification techniques
• Superior performance on benchmark instances